# Cosmology - an expansion of all length scales

From the link Is non-mainstream physics appropriate for this site?

"a question that proposes a new concept or paradigm, but asks for evaluation of that concept within the framework of current (mainstream) physics is OK."

Here is a concept, evaluation within the framework of current (mainstream) physics would be welcome.

Is it possible that an expansion of all length scales can be happening, as in the cartoon below?

It shows all lengths increasing, the size of atoms, people, stars and the distances between all objects. Each physical quantity and constant varies depending on the number of length dimensions in it. For example since Planck's constant has a length dimension of 2, so it's change with time is

$$h=h_0e^{2Ht}$$

where $$H$$ is an expansion constant and $$t$$ is time.

$$\begin{array}{c|c|c} {quantity} & {length-dimension} & {change}\\ \hline length & 1 & e^{Ht}\\ mass & 0 & constant\\ time & 0 & constant\\ h & 2 & e^{2Ht}\\ c & 1 & e^{Ht}\\ G & 3 & e^{3Ht}\\ Area & 2 & e^{2Ht}\\ \end{array}$$

etc...

Can this type of expansion be ruled out A) locally or B) by distant measurements e.g. of distant stars or galaxies, from within mainstream physics?

The expansion referred to occurs for the whole universe. It's proposed as there could be another reason for the redshift of light from distant stars. If the energy of a photon is conserved during flight, but was emitted when Planck's constant was lower, then from $$E=hf$$, the frequency of the received photon would be lower and the light from a distant star would be redshifted.

A bounty has now been added. A convincing reason why the above type of expansion cannot be occurring would be welcome.

Here is the work done so far.

It is to determine the apparent matter density that would be concluded in a flat universe, with a matter density of $$1.0$$ and the type of expansion above.

It leads to the conclusion that the matter density would be measured to be $$0.25$$ or $$0.33$$ from galaxy clusters and supernovae data respectively. A Diagram of supernovae data is below and then more details of the calculations.

and

The diagrams show the distance modulus predicited by the type of expansion in the question, top curve. Concordance cosmology with a matter density of 0.3 and 1.0 are the middle and bottom curve respectively. The second diagram is an enlargement of the first.

Matter density from Galaxy Clusters etc...

Traditionally the scale factor of the universe at redshift $$z$$ is

$$a=\frac{1}{1+z}\tag{1}$$

If the energy of the photon is conserved during flight, from $$E=hf$$ and $$h=h_0e^{2Ht}$$

For an emitted wavelength of $$\lambda_1$$

$$z=\frac{\lambda_1e^{2Ht}-\lambda_1}{\lambda_1}$$

$$1+z = e^{2Ht}=a^{-2}$$ ,

($$a$$ decreases with increasing $$z$$ in an expanding universe) so

$$a=\frac{1}{\sqrt{1+z}}\tag{2}$$

For small distance $$d$$

$$\frac{v}{c} =z= e^{2H\frac{d}{c}}-1=\frac{2Hd}{c}$$

$$v=2Hd\tag{3}$$

i.e. Hubble’s law is still valid but we identify the expansion parameter $$H$$ with half of Hubble’s constant $$H_0$$

this leads to the conclusion that the matter density will be measured to be $$\frac{1}{4}$$ of the true value, as follows.

$$\Omega_m = \frac{\rho}{\rho_{crit}}\tag{4}$$

$$\rho_{crit}=\frac{3H(z)^2}{8\pi G}\tag{5}$$

If the value for $$H(z)$$ used in $$\rho_{crit}$$ is twice the true value, then the apparent matter density would be measured as $$0.25$$ instead of $$1$$.

Matter Density from Supernovae Data.

In LCDM the Hubble parameter is

$$H(z)=H_0\sqrt{\Omega_m {(1+z)}^3+\Omega_k{(1+z)}^2+\Omega_\Lambda}$$

The comoving distance is obtained from

$$D_M=\int_0^z \frac{c}{H(z)} dz$$

Using a flat universe approximation, omitting $$\frac{c}{H_0}$$ and using $$m$$ for $$\Omega_m$$ ,the comoving distance, for small $$z$$ is

$$\int_0^z(m(1+3z+3z^2+\dots )+1-m)^{-\frac{1}{2}}dz$$

$$=\int_0^z(1+3mz+3mz^2)^{-\frac{1}{2}}dz =\int_0^z(1-\frac{3}{2}mz+\dots)dz$$

$$=z-\frac{3mz^2}{4}\tag{6}$$

For the type of expansion that we hope to rule out,

The co-moving distance is

$$D_M=\int_t^0 \frac{c}{a(t)} dt$$

$$a=\frac{1}{\sqrt{1+z}}$$

$$\frac{da}{dt}=\frac{da}{dz} \times \frac{dz}{dt} ={-\frac{1}{2}(1+z)^{-\frac{3}{2}}}\times\frac{dz}{dt}$$

$$H(z)=H=\frac{\dot{a}}{a}=\frac{-1}{2(1+z)}\times\frac{dz}{dt}$$

$$dt=\frac{-1}{2H(1+z)}dz$$

$$D_M=\int_0^z \frac{c}{2H}{(1+z)}^{-\frac{1}{2}} dz$$

$$D_M=\frac{2c}{H_0}(\sqrt{1+z}-1)\tag{7}$$

again omitting $$\frac{c}{H_0}$$ and for small $$z$$, $$(7)$$ becomes

$$2(1+\frac{1}{2}z-\frac{1}{8}z^2-1)$$

$$=z-\frac{z^2}{4}\tag{8}$$

there is a match between $$(6)$$ and $$(8)$$ if $$m=\frac{1}{3}$$

So we conclude from Galaxy and supernovae data, or combinations of data sets, that the matter density would be measured, with the type of expansion in the question, at between $$0.25$$ and $$0.33$$. As it is measured at this value, it's concluded that the expansion cannot be ruled out this way. A diagram with supernovae data is above.

Is there a convincing reason why the expansion described should be ruled out?

• The gauge principle originally came from Weyl postulating such a scale invariance inspired from the principle of relativity. Weyl showed his work to Einstein who said it was more mathematics than physics. It took another fifty years before it found its correct formulation ... Mar 13, 2021 at 10:46
• That's interesting, Einstein's comment that it was mathematical sounds as though he thought it had no meaning physically, it can easily be assumed that such an expansion is meaningless as no change can ever be measurable. But can a physical meaning be found if we compare length scales here to those far away, i.e. in a totally static universe they would be the same, but in such an expanding one, they are larger now than they were before and larger than they were when light left a distant star. Have any cosmological models been developed incorporating the type of expansion described? Mar 13, 2021 at 10:54
• Related, and links therein.
– rob
Mar 13, 2021 at 11:13
• There have been articles wondering something similar, if we also identify the expansion constant with the Hubble parameter, then since the redshift depends on $2H$ then the true 'expansion parameter' is half of the Hubble parameter. It has the advantage that when we work out the matter density from $\frac{\rho}{\rho_{crit}}$ since the denominator depends on $H^2$ it works out as 0.25 and may account for the apparent dark energy phenomenon Mar 13, 2021 at 11:20
• @John Hunter: Thats roughly right. Weyl was looking for a local invariance of scale. Gauge, in one sense of the word means scale. So your non-mainstream example is very mainstream, it just happens to be carefully hidden away under many layers of technical jargon! Mar 13, 2021 at 11:33

## 5 Answers

Mainstream physics/cosmology says that local systems that are held together e.g. by gravity or electromagnetic forces do not take part in the global expansion. Our solar system had the same size billions of years ago (there is certainly no evidence to the contrary) and atoms in galaxies billions of light years (in space and time) away have the same size as those locally (as one can conclude from the spectra of distant objects).

See also this reference https://arxiv.org/abs/gr-qc/0508052

Anyway, if your ruler expands as well (like you have drawn above) there would not be an expansion of the universe in the first place, as you would always measure the same distance to a galaxy.

• but when we look at the spectra of distant objects, they are redshifted and the photons have a reduced energy. This would be what we would see if the atoms in the distant object were smaller than the same type of atom today. Mar 13, 2021 at 17:06
• @JohnHunter Smaller atoms would have a higher (negative) energy (electrons are closer to the nucleus) so they would emit a blue-shifted spectrum. But if you subtract the redshift, the frequencies are not blue-shifted. See also this reference arxiv.org/pdf/gr-qc/0508052.pdf Mar 13, 2021 at 17:21
• we have to include everything i.e. there are formulae for energy levels e.g. $E=\frac{-Zk}{r}$ but when we look at the units of $k$, related to the Coulomb constant, it has units of $[L^3]$, the overall effect is that the energy of an emitted photon is proportional to $L^2$, which it has to be, as it's an energy. The emitted photons from the smaller atom would have less energy and be redshifted Mar 13, 2021 at 17:47
• @JohnHunter The Coulomb constant is just that, a constant. It does not depend on the distance of the charges (by the way, you forgot the charges in your equation for the energy). The gravitational constant is not changed either by the expansion of the universe (there have been some (non-mainstream) theories that propose a variable G but there is no observational evidence for this) Mar 13, 2021 at 18:14
• The gravitational constant is also not changed by the expansion in the question, not in a measurable way anyway, so is not ruled out by Lunar Laser Ranging for example. You're point that mainstream physics says that local systems are held together by gravity and do not take part in the expansion is true (i.e. it does say that). Mar 17, 2021 at 15:28

the frequency of the received photon would be lower

Why would it ? Since $$c=\lambda f$$ and $$c$$ and $$\lambda$$ change in the same proportion then $$f$$ is constant. All you are doing is changing the units in which length is measured. You get exactly the same effect if you measure the wavelength in furlongs instead of metres, and denominate the speed of light in furlongs per second - frequency remains unchanged.

• It's different to a change of units. The energy of the emitted photon was $E=hf$ at a time when both the energy of the photon and Planck's constant were lower. Then when the photon arrives here Planck's constant has increased. If the energy of the photon is conserved during it's journey, then the measured frequency would be reduced. Mar 13, 2021 at 16:28
• @JohnHunter But if the length scale changes then units of energy also change since dimensions of energy are $ML^2T^{-2}$. So the value of $E$ changes numerically in the same proportion as $h$. Once again, $f$ is unaffected. If you insist on some new principle that means that $E$ does not change numerically if the length scale changes then you have moved away from mainstream physics. Mar 13, 2021 at 16:37
• The energy of the photon is conserved as no time passes for it. The $t$ in the equation is the time that is observed to pass for the object. Can you think of a way either locally or with measurements at a distance to rule out such an 'expansion'? Mar 13, 2021 at 16:42
• @JohnHunter If you change the length scale uniformly for everything then the laws of physics are unchanged, so no measurement would detect this change. But this is trivial - it is just the same as if I use metres as my unit of length today and furlongs tomorrow. You seem to be trying to make a change of units sound profound. I am done here. Mar 13, 2021 at 17:29
• @JohnHunter: but according to your assumptions, the shoelace should have grown as well (in the change of length scale case), so you would not notice the change in length scale. However, if your friend has performed quality control by measuring the lowest resonant frequency of light being reflected back and forth between the ends of the shoelace, and he tells you the number, you will notice that the frequency was higher in the past (because the shoelace was smaller then) than what you measure for the showlace after it has arrived. Mar 17, 2021 at 11:17

I ll just simply write couple of reasons why its not possible.

1. If you are talking about everything getting bigger in size then the measurement devices (such as rulers, etc) will also get bigger at the same amount. So even that is the point theres no real way of measuring it. So its not reasonable to talk about it. I can also argue that everything is getting smaller ? Can you argue that as well ?

2. The force between the two electrons is about $$10^{40}$$ times larger than the gravitational force. If the expansion of the universe does not have any effect in our solar system (which is governed by gravitational force), then clearly (and logically) the expansion of the universe cannot have any effect on the atomic-scales.

3. From a simplest point of view, if just my size increases but my mass stays the same my density must get lower and lower which is not the case for me or for any other object that is around you.

4. Occam's razor - Why everything should be expanded at the same amount in the first place ? Whats the point ?

• 1. Observational evidence (assuming the type of expansion in the question) is that H is positive. 2. The type of expansion in the question does affect the size of the solar system. 3. How would you measure density?, if you did it by seeing if an object floats or sinks in a liquid, the density of the liquid would have changed too. If it's from $D=M/V$, the lengths to measure $V$ would seem unchanged as your ruler has changed too. 4. Occam's razor, yes, it's simplest that everything expands equally, why should the distance between galaxies expand but other objects not! Mar 17, 2021 at 12:15
• @JohnHunter@I am saying that if everything expands at the same amount you cannot detect the expansion. That's actually the proof that why we can measure that the universe is expanding. Everything is expanding at the same rate (for instance $H$) we would not be able to observe that. 2) No. It does not effect. 3) I am not talking about the measurements but the "physical effects" of being less dense. 4) Okay I am claiming that everything is getting smaller by the same amount argue that please Mar 17, 2021 at 12:29
• It's a good point that everything could be getting smaller. then $H$ would be negative, to regain symmetry we could say that it happens in an antimatter universe in which time is negative too, then $-H\times-t$ becomes positive. This could be the reason why we seem to live in a matter universe, i.e. if time was reversed matter would be antimatter, but still perceived as matter, $H$ and $t$ would become negative but still perceived as positive. Mar 17, 2021 at 15:01

If all things expand, atoms become bigger. Bigger atoms have at least different electromagnetic spectra. Hence, if we look at the hydrogen lines from the andromeda galaxy, we see them like they were 2.5 million years ago, and that would mean, emitted by much smaller atoms. The energy levels of the hydrogen atom are proportional to the reciprocal of the Bohr radius $$a_0$$ and if this was smaller in the past, that means higher energy differences, or higher frequencies.

Of course the wavelength of the received light would have been expanded since then, due to the alleged spatial expansion, but their frequency would have stayed the same, because time is not affected by the expansion, as I understand you.

Hence, we should see different atomic spectra for distant objects, namely blue-shifted ones (higher atomic frequencies in the past), which has not been observed, as to my knowledge.

• It's true that the Bohr radius would be lower in the past, but (see Thomas' answer), we have to include everything i.e. there are formulae for energy levels e.g. $E=−Zke^2/r$ but when we look at the units of $k$, related to the Coulomb constant, it has units of $L^3$, the overall effect is that the energy of an emitted photon is proportional to $L^2$, which it has to be, as it's an energy. The emitted photons from the smaller atom would have less energy and be redshifted Mar 17, 2021 at 12:07
• I am getting the impression that your model is not so well-defined with respect to what changes and what stays constant. Time has also dimension of length. Actually the speed of light is just a definition as of today. So how could space expand without affecting time as well? So this seems to boil down to gandalf61's answer, that you just assume different measurement units over time. Mar 17, 2021 at 12:17
• as we are discussing it in gandalf61's answer, a comment will be posted there... Mar 17, 2021 at 22:31

It is hard to write a precise answer because it is not clear to me that the concept being presented is sufficiently worked out to make something one can assess. This may be a failure of understanding on my part, but I hope it will help if I simply give a few reactions so that you can see what a physicist with reasonable general knowledge of this area makes of this (but I am not a cosmologist).

At the start one first suspects that the question is asking about something unobservable but I think it is not doing that. It is proposing observable things. But it is not clear to me that it hangs together. For example, the question assumes that the combination $$h f$$ for a photon is conserved as it propagates long distances, and if $$E$$ is constant and $$h$$ varies as $$h = h_0 e^{2 Ht}$$ then one has $$f = f_0 e^{-2 H t}.$$ Meanwhile the question also proposes $$c = c_0 e^{Ht}$$ so this gives, for the wavelength, $$\lambda = \frac{c}{f} = \frac{h_0 c_0}{E} e^{3 Ht}.$$

The next equation in the question reads $$z \stackrel{?}{=} \frac{\lambda_1 e^{2 Ht} - \lambda_1}{\lambda_1}$$ where I have put a query in order to signal that I am not sure where this equation came from since I was expecting $$e^{3 Ht}$$. I suppose what might have happened is that $$z$$ was being defined as $$z \equiv (f_{r}^{-1} - f_{e}^{-1})/f_e^{-1}$$, where the subscripts stand for 'received' and 'emitted', so then one has $$z \equiv \frac{f_{r}^{-1} - f_{e}^{-1}}{f_e^{-1}} = \frac{f_e}{f_r} - 1 = e^{2 Ht} - 1$$ and the rest of the question assumes this result.

The above reveals one difficulty I have with the whole approach. It is not clear to me that it can hang together overall. When we detect the light from distant supernovae, the instruments to measure red shift use, I think, optical methods such as diffraction gratings and Michelson interferometers, so it is wavelength not frequency that they are measuring. But when they detect luminosity then it is, I think, more like an energy measurement. Meanwhile, when we do stellar physics calculations to describe the supernovae (or other stars) we adopt the standard methods and mostly think in terms of energy and frequency. So if frequency and wavelength are scaling differently as cosmological time goes on, we don't have, at the outset, any idea of which of our calculations to trust, or which aspect of them, until a lot more working out has been done. And the suspicion is that that working-out cannot be done because it won't in fact hang together as a logical whole. I am not being so bold as to assert that; I am simply giving a response which signals some of the issues which need to be addressed before the ideas can be taken forward.

• I am glad that I am not the only one having a hard time in making sense of this. Generally, I would say it is primarily the responsibility of the author to express his ideas coherently. If the deviation from mainstream is greater, stackexchange may not be the right place to discuss it because there is too much to explain from the side of the author. Mar 17, 2021 at 12:27
• It's a good answer, and true that it might not hang together as a 'logical whole', although efforts have been made to make it do so - maybe answers will show whether it does or not. For the equation where you've put a ? $\lambda_e=c_e/f_e$ and $\lambda_r=c_r/f_r$ , so $\lambda_r=c_ee^{Ht}/f_ee^{-2Ht} = \lambda_e e^{3Ht}$ bu the measurable change is reduced by another factor,as rulers etc...on arrival are larger than when emitted, so it's back to $\lambda_e e^{2Ht}$ Mar 17, 2021 at 13:22
• @JohnHunter supposing one accepts that for the sake of argument, the next difficulty is that you freely quote various results from LCDM cosmology but without showing that the reasoning leading to those results still holds. Mar 17, 2021 at 15:36
• well, the post had to be kept reasonably short and readable, are there any particular results or equations that you think may not be valid? Mar 18, 2021 at 10:29